Normal form transformations for modulated deep-water gravity waves
DOI:
https://doi.org/10.33044/revuma.2918Abstract
Modulation theory is a well-known tool to describe the long-time evolution and stability of small-amplitude, oscillating solutions to dispersive nonlinear partial differential equations. There have been a number of approaches to deriving envelope equations for weakly nonlinear waves. Here we review a systematic method based on Hamiltonian transformation theory and averaging Hamiltonians. In the context of the modulation of two- or three-dimensional deep-water surface waves, this approach leads to a Dysthe equation that preserves the Hamiltonian character of the water wave problem. An explicit calculation of the third-order Birkhoff normal form that eliminates all non-resonant cubic terms yields a non-perturbative procedure for the reconstruction of the free surface. We also present new numerical simulations of this weakly nonlinear approximation using the version with exact linear dispersion. We compare them against computations from the full water wave system and find very good agreement.
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